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a(n) = binomial(n-1,3)*n^(n-4).
5

%I #20 Sep 08 2022 08:45:00

%S 0,0,0,1,20,360,6860,143360,3306744,84000000,2338460520,70946979840,

%T 2332989862060,82726831323136,3148511132812500,128071114403348480,

%U 5546563698427324720,254873089955815096320,12387799656377835411984

%N a(n) = binomial(n-1,3)*n^(n-4).

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Prop. 5.3.2.

%H G. C. Greubel, <a href="/A053508/b053508.txt">Table of n, a(n) for n = 1..250</a>

%F E.g.f.: LambertW(-x)^4/4!. - _Vladeta Jovovic_, Apr 07 2001

%t Table[Binomial[n-1,3]n^(n-4),{n,25}] (* _Harvey P. Dale_, Jun 17 2014 *)

%t With[{nmax = 25}, CoefficientList[Series[LambertW[-x]^4/4!, {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Nov 14 2017 *)

%o (PARI) vector(25, n, binomial(n-1,3)*n^(n-4)) \\ _G. C. Greubel_, Jan 18 2017

%o (Magma) [Binomial(n-1,3)*n^(n-4): n in [1..25]]; // _G. C. Greubel_, Nov 14 2017

%o (Sage) [binomial(n-1,3)*n^(n-4) for n in (1..25)] # _G. C. Greubel_, May 15 2019

%o (GAP) List([1..25], n-> Binomial(n-1,3)*n^(n-4)) # _G. C. Greubel_, May 15 2019

%Y Cf. A000169, A053506, A053507, A053509.

%K nonn

%O 1,5

%A _N. J. A. Sloane_, Jan 15 2000